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Question
It is observed that it rains on 12 days out of 30 days. Find the probability that it rains exactly 3 days of week.
Solution
Let X = number of days it rains in a week.
p = probability that it rains
∴ p = `12/30 = 2/5`
and q = 1 - p = `1 - 2/5 = 3/5`
Given: n = 7
∴ X ~ B `(7, 2/5)`
The p.m.f. of X is given by
P(X = x) = `"^nC_x p^x q^(n - x)`
i.e. p(x) = `"^7C_x (2/5)^x (3/5)^(7 - x)` x = 0, 1, 2, ...., 7
P(it rains exactly 3 days of week) = P(X = 3)
= p(3) = `"^7C_3 (2/5)^3 (3/5)^(7 - 3)`
`= (7 xx 6 xx 5)/(3 xx 2 xx 1) (8/125)(81/625)`
`= 35(8/125)(81/625) = (35 xx 8 xx 81)/5^7`
`= 22680/78125 = 0.2903`
Hence, the probability that it rains exactly 3 days of week = `35 xx 8 xx 81/5^7` OR 0.2903.
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